Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains

P. Poláčik, Susanna Terracini

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We consider a semilinear elliptic equation on a smooth bounded domain Ω in ℝ2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal is to exhibit examples of equations which admit nonnegative, nonzero solutions for which the second property fails; necessarily, such solutions have a nontrivial nodal set in Ω. Previously, such examples were known for nonsmooth domains only.

Original languageEnglish (US)
Pages (from-to)1249-1259
Number of pages11
JournalProceedings of the American Mathematical Society
Volume142
Issue number4
DOIs
StatePublished - Feb 10 2014

Keywords

  • Nodal set
  • Nonnegative solutions
  • Planar domain
  • Semilinear elliptic equation

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